Thrust Vector Control of an Upper-Stage Rocket with Multiple Propellant Slosh Modes

Abstract

The thrust vector control problem for an upper-stage rocket with propellant slosh dynamics is considered. The control inputs are defined by the gimbal deflection angle of a main engine and a pitching moment about the center of mass of the spacecraft. The rocket acceleration due to the main engine thrust is assumed to be large enough so that surface tension forces do not significantly affect the propellant motion during main engine burns. A multi-mass-spring model of the sloshing fuel is introduced to represent the prominent sloshing modes. A nonlinear feedback controller is designed to control the translational velocity vector and the attitude of the spacecraft, while suppressing the sloshing modes. The effectiveness of the controller is illustrated through a simulation example.

1. Introduction

In fluid mechanics, liquid slosh refers to the movement of liquid inside an accelerating tank or container. Important examples include propellant slosh in spacecraft tanks and rockets (especially upper stages), cargo slosh in ships and trucks transporting liquids, and liquid slosh in robotically controlled moving containers.

A variety of passive methods have been employed to mitigate the adverse effect of sloshing, such as introducing baffles or partitions inside the tanks [1, 2]. These techniques do not completely succeed in canceling the sloshing effects. Thus, active control methods have been proposed for the suppression of sloshing effectively.

The control approaches developed for robotic systems moving liquid filled containers [3–11] and for accelerating space vehicles are mostly based on linear control design methods [12, 13] and adaptive control methods [14]. The linear control laws for the suppression of the slosh dynamics inevitably lead to excitation of the transverse vehicle motion through coupling effects. The complete nonlinear dynamics formulation in this paper allows simultaneous control of the transverse, pitch, and slosh dynamics.

Some of the control design methods use command input shaping methods that do not require sensor measurements of the sloshing liquid [3, 8–11], while others require either sensor measurements [4, 5] or observer estimates of the slosh states [15]. In most of these approaches, only the first sloshing mode represented by a single pendulum model or a single mass-spring model has been considered and higher slosh modes have been ignored. The literature that considers more than one sloshing mode in modeling the slosh dynamics includes [2, 16].

In this paper, a mechanical-analogy model is developed to characterize the propellant sloshing during a typical thrust vector control maneuver. The spacecraft acceleration due to the main engine thrust is assumed to be large enough so that surface tension forces do not significantly affect the propellant motion during main engine burns. This situation corresponds to a “high-𝑔’’ (here 𝑔 refers to spacecraft acceleration) regime that can be characterized by using the Bond number Bo—the ratio of acceleration-related forces to the liquid propellants surface tension forces, which is given by Bo=𝜌𝑎𝑅²𝜎,(1.1) where 𝜌 and 𝜎 denote the liquid propellants density and surface tension, respectively, 𝑎 is the spacecraft acceleration, and 𝑅 is a characteristic dimension (e.g., propellant tank radius). During the steady-state high-𝑔 situation, the propellant settles at the “bottom’’ of the tank with a flat free surface. When the main engine operation for thrust vector control introduces lateral accelerations, the propellant begins sloshing. As discussed in [17], Bond numbers as low as 100 would indicate that low-gravity effects may be of some significance. A detailed discussion of low-gravity fluid mechanics is given in [2].

The previous work in [16] considered a spacecraft with multiple fuel slosh modes assuming constant physical parameters. In this paper, these results are extended to account for the time-varying nature of the slosh parameters, which renders stability analysis more difficult. The control inputs are defined by the gimbal deflection angle of a nonthrottable thrust engine and a pitching moment about the center of mass of the spacecraft. The control objective, as is typical in a thrust vector control design for a liquid upper stage spacecraft during orbital maneuvers, is to control the translational velocity vector and the attitude of the spacecraft, while attenuating the sloshing modes characterizing the internal dynamics. The results are applied to the AVUM upper stage—the fourth stage of the European launcher Vega [18]. The main contributions in this paper are (i) the development of a full nonlinear mathematical model with time-varying slosh parameters and (ii) the design of a nonlinear time-varying feedback controller. A simulation example is included to illustrate the effectiveness of the controller.

2. Mathematical Model

This section formulates the dynamics of a spacecraft with a single propellant tank including the prominent fuel slosh modes. The spacecraft is represented as a rigid body (base body) and the sloshing fuel masses as internal bodies. In this paper, a Newtonian formulation is employed to express the equations of motion in terms of the spacecraft translational velocity vector, the angular velocity, and the internal (shape) coordinates representing the slosh modes. A multi-mass-spring model is derived for the sloshing fuel, where the oscillation frequencies of the mass-spring elements represent the prominent sloshing modes [19].

Consider a rigid spacecraft moving on a plane as indicated in Figure 1, where 𝑣_𝑥,𝑣_𝑧 are the axial and transverse components, respectively, of the velocity of the center of the fuel tank, and 𝜃 denotes the attitude angle of the spacecraft with respect to a fixed reference. The fluid is modeled by moment of inertia 𝐼₀ assigned to a rigidly attached mass 𝑚₀ and point masses 𝑚_𝑖,𝑖=1,…,𝑁, whose relative positions along the spacecraft fixed 𝑧-axis are denoted by 𝑠_𝑖. Moments of inertia 𝐼_𝑖 of these masses are usually negligible. The locations ℎ₀ and ℎ_𝑖 are referenced to the center of the tank. A restoring force −𝑘_𝑖𝑠_𝑖 acts on the mass 𝑚_𝑖 whenever the mass is displaced from its neutral position 𝑠_𝑖=0. A thrust 𝐹 is produced by a gimballed thrust engine as shown in Figure 1, where 𝛿 denotes the gimbal deflection angle, which is considered as one of the control inputs. A pitching moment 𝑀 is also available for control purposes. The constants in the problem are the spacecraft mass 𝑚 and moment of inertia 𝐼, the distance 𝑏 between the body 𝑧-axis and the spacecraft center of mass location along the longitudinal axis, and the distance 𝑑 from the gimbal pivot to the spacecraft center of mass. If the tank center is in front of the spacecraft center of mass, then 𝑏>0. The parameters 𝑚₀,𝑚_𝑖,ℎ₀,ℎ_𝑖,𝑘_𝑖, and 𝐼₀ depend on the shape of the fuel tank, the characteristics of the fuel and the fill ratio of the fuel tank. Note that these parameters are time-varying, which renders the Lyapunov-based stability analysis more difficult.

Figure 1 

A multiple-slosh mass-spring model for a spacecraft.

To preserve the static properties of the liquid, the sum of all the masses must be the same as the fuel mass 𝑚_𝑓, and the center of mass of the model must be at the same elevation as that of the fuel, that is, 𝑚₀+^𝑁_𝑖=1𝑚₀𝑚_𝑖=𝑚_𝑓,ℎ₀+^𝑁_𝑖=1𝑚_𝑖ℎ_𝑖=0.(2.1) Assuming a constant fuel burn rate, we have 𝑚_𝑓=𝑚_ini1−𝑡𝑡_𝑓,(2.2) where 𝑚_ini is the initial fuel mass in the tank and 𝑡_𝑓 is the time at which, at a constant rate, all the fuel is burned.

To compute the slosh parameters, a simple equivalent cylindrical tank is considered together with the model described in [2], which can be summarized as follows. Assuming a constant propellant density, the height of still liquid inside the cylindrical tank is ℎ=4𝑚^𝑓𝜋𝜑²𝜌,(2.3) where 𝜑 and 𝜌 denote the diameter of the tank and the propellant density, respectively. As shown in [2], every slosh mode is defined by the parameters 𝑚_𝑖=𝑚_𝑓𝜑tanh2𝜉^𝑖ℎ/𝜑ℎ𝜉_𝑖𝜉^2𝑖−1ℎ,(2.4)_𝑖=ℎ2−2𝜑𝜉_𝑖tanh𝜉^𝑖ℎ𝜑−1−cosh2𝜉^𝑖ℎ/𝜑𝑘sinh2𝜉_𝑖ℎ/𝜑,(2.5)_𝑖=𝑚^𝑖𝑔𝜑2𝜉^𝑖tanh2𝜉^𝑖ℎ𝜑,(2.6) where 𝜉_𝑖, for all 𝑖, are constant parameters given by 𝜉₁=1.841,𝜉₂=5.329,𝜉_𝑖≃𝜉_𝑖−1+𝜋,(2.7) and 𝑔 is the axial acceleration of the spacecraft. For the rigidly attached mass, 𝑚₀ and ℎ₀ are obtained from (2.1)–(2.5). Assuming that the liquid depth ratio for the cylindrical tank (i.e., ℎ/𝜑) is less than two, the following relations apply: 𝐼₀=1−0.85ℎ𝜑𝑚^𝑓31𝜑6²+1ℎ2²−𝑚⁰ℎ²⁰−^𝑁_𝑖=1𝑚_𝑖ℎ^2𝑖,ifℎ𝐼𝜑<1,₀=0.35ℎ𝜑−0.2𝑚^𝑓31𝜑6²+1ℎ2²−𝑚⁰ℎ²⁰−^𝑁_𝑖=1𝑚_𝑖ℎ^2𝑖,if1≤ℎ𝜑<2.(2.8)

Let 𝑖̂ and 𝑘̂ be the unit vectors along the spacecraft-fixed longitudinal and transverse axes, respectively, and denote by (𝑥,𝑧) the inertial position of the center of the fuel tank. The position vector of the center of mass of the vehicle can then be expressed in the spacecraft-fixed coordinate frame as ^→𝑟=(𝑥−𝑏)𝑖̂+𝑧𝑘̂.(2.9) The inertial velocity and acceleration of the vehicle can be computed as ̇‌_→̈𝑟=𝑣_𝑥𝑖̂+𝑣_𝑧+𝑏𝜃̇𝑘̂,_→𝑟=𝑎_𝑥+𝑏𝜃̇²𝑖̂+𝑎^𝑧+𝑏𝜃̈𝑘̂,(2.10) where we have used the fact that (𝑣_𝑥,𝑣_𝑧)=(𝑥̇+𝑧𝜃̇,𝑧̇−𝑥𝜃̇) and (𝑎_𝑥,𝑎_𝑧)=(𝑣̇_𝑥+𝑣_𝑧𝜃̇,𝑣̇_𝑧−𝑣_𝑥𝜃̇).

Similarly, the position vectors of the fuel masses 𝑚₀,𝑚_𝑖, for all 𝑖, in the spacecraft-fixed coordinate frame are given, respectively, by ^→𝑟₀=𝑥+ℎ₀𝑖̂+𝑧𝑘̂,_→𝑟_𝑖=𝑥+ℎ_𝑖𝑖̂+𝑧+𝑠_𝑖𝑘̂,∀𝑖.(2.11)

The inertial accelerations of the fuel masses can be computed as ̈‌_→̈𝑟₀=𝑎_𝑥−ℎ₀𝜃̇²+ℎ̈₀𝑖̂+𝑎^𝑧−2ℎ̇₀𝜃̇−ℎ₀𝜃̈𝑘̂,_→𝑟_𝑖=𝑎_𝑥+𝑠_𝑖𝜃̈−ℎ_𝑖𝜃̇²+ℎ̈_𝑖+2𝑠_𝑖̇𝜃̇𝑖̂+𝑎_𝑧+𝑠_𝑖̈−ℎ_𝑖𝜃̈−𝑠_𝑖𝜃̇²−2ℎ̇_𝑖𝜃̇𝑘̂,∀𝑖.(2.12)

Now Newton’s second law for the whole system can be written as 𝐹^→=𝑚^→𝑟̈+^𝑁_𝑖=0𝑚_𝑖̈‌_→𝑟_𝑖,(2.13) where 𝐹^→=𝐹𝑖̂cos𝛿+𝑘̂sin𝛿.(2.14)

The total torque with respect to the tank center can be expressed as 𝜏=^→𝐼+𝐼₀+^𝑁_𝑖=1𝐼_𝑖𝜃̈𝑗+̂^→𝜌×𝑚^→𝑟̈+^𝑁_→𝑖=0𝜌_𝑖×𝑚^→𝑟̈_𝑖,(2.15) where 𝜏=𝜏𝑗̂^→=[𝑀+𝐹(𝑏+𝑑)sin𝛿]𝑗,̂(2.16) and ^→𝜌,^→𝜌₀, and ^→𝜌_𝑖 are the positions of 𝑚,𝑚₀, and 𝑚_𝑖 relative to the tank center, respectively, that is, 𝜌=−𝑏𝑖,̂^→→𝜌₀=ℎ₀𝑖,̂^→𝜌_𝑖=ℎ_𝑖𝑖̂+𝑠_𝑖𝑘̂,∀𝑖.(2.17)

The dissipative effects due to fuel slosh are included via damping constants 𝑐_𝑖. When the damping is small, it can be represented accurately by equivalent linear viscous damping. Newton’s second law for the fuel mass 𝑚_𝑖 can be written as 𝑚_𝑖𝑎_𝑧𝑖=−𝑐_𝑖𝑠_𝑖̇−𝑘_𝑖𝑠_𝑖,(2.18) where 𝑎_𝑧𝑖=𝑠_𝑖̈+𝑎_𝑧−ℎ_𝑖𝜃̈−𝑠_𝑖𝜃̇²−2ℎ̇_𝑖𝜃̇.(2.19)

Using (2.13)–(2.18), the equations of motion can be obtained as 𝑚𝑚++𝑚𝑚_𝑓𝑓𝑎_𝑥+𝑚𝑏𝜃̇²+^𝑁_𝑖=1𝑚_𝑖𝑠_𝑖𝜃̈+2𝑠_𝑖̇𝜃̇+ℎ̈_𝑖+𝑚₀ℎ̈₀=𝐹cos𝛿,(2.20)𝐼𝜃̈+_𝑁𝑎_𝑧+𝑚𝑏𝜃̈+^𝑁_𝑖=1𝑚_𝑖𝑠^𝑖̈−𝑠_𝑖𝜃̇²−2ℎ̇_𝑖𝜃̇−2𝑚₀ℎ̇₀𝜃̇=𝐹sin𝛿,(2.21)_𝑖=1𝑚_𝑖𝑚𝑠_𝑖𝑎_𝑥−ℎ_𝑖𝑠_𝑖̈+2𝑠_𝑖𝑠_𝑖̇+ℎ_𝑖ℎ̇_𝑖𝜃̇+𝑠_𝑖ℎ̈_𝑖+2𝑚₀ℎ₀ℎ̇₀𝜃̇₀+𝑚𝑏𝑎_𝑧=𝜏,(2.22)_𝑖𝑠^𝑖̈+𝑎_𝑧−ℎ_𝑖𝜃̈−𝑠_𝑖𝜃̇²−2ℎ̇_𝑖𝜃̇+𝑘_𝑖𝑠_𝑖+𝑐_𝑖𝑠_𝑖̇=0,∀𝑖,(2.23) where 𝑝=𝑏+𝑑 and 𝐼=𝐼+𝐼₀+𝑚𝑏²+𝑚₀ℎ²⁰+^𝑁_𝑖=1𝐼_𝑖+𝑚_𝑖ℎ^2𝑖+𝑠^2𝑖.(2.24)

The control objective is to design feedback controllers so that the controlled spacecraft accomplishes a given planar maneuver, that is a change in the translational velocity vector and the attitude of the spacecraft, while suppressing the fuel slosh modes. Equations (2.20)–(2.23) model interesting examples of underactuated mechanical systems. The published literature on the dynamics and control of such systems includes the development of theoretical controllability and stabilizability results for a large class of systems using tools from nonlinear control theory and the development of effective nonlinear control design methodologies [20] that are applied to several practical examples, including underactuated space vehicles [21, 22] and underactuated manipulators [23].

3. Nonlinear Feedback Controller

This section presents a detailed development of feedback control laws through the model obtained via the multi-mass-spring analogy.

Consider the model of a spacecraft with a gimballed thrust engine shown in Figure 1. If the thrust 𝐹 during the fuel burn is a positive constant, and if the gimbal deflection angle and pitching moment are zero, 𝛿=𝑀=0, then the spacecraft and fuel slosh dynamics have a relative equilibrium defined by 𝑣_𝑧=𝑣_𝑧,𝜃=𝜃,𝜃̇=0,𝑠_𝑖=0,𝑠_𝑖̇=0,∀𝑖,(3.1) where 𝑣_𝑧 and 𝜃 are arbitrary constants. Without loss of generality, the subsequent analysis considers the relative equilibrium at the origin, that is, 𝑣_𝑧=𝜃=0. Note that the relative equilibrium corresponds to the vehicle axial velocity 𝑣_𝑥(𝑡)=𝑣_𝑥0+𝑎_𝑥𝑡,𝑡≤𝑡_𝑏,(3.2) where 𝑣_𝑥0 is the initial axial velocity of the spacecraft, 𝑡_𝑏 is the fuel burn time, and 𝑎_𝑥=𝐹𝑚+𝑚_𝑓.(3.3)

Now assume the axial acceleration term 𝑎_𝑥 is not significantly affected by small gimbal deflections, pitch changes, and fuel motion (an assumption verified in simulations). Consequently, (2.20) becomes 𝑣̇_𝑥+𝜃̇𝑣_𝑧=𝑎_𝑥.(3.4) Substituting this approximation leads to the following reduced equations of motion for the transverse, pitch, and slosh dynamics: 𝐼𝜃̈+𝑚_𝑁+𝑚_𝑓𝑎_𝑧̂+𝑚𝑏𝜃̈+^𝑁_𝑖=1𝑚_𝑖𝑠^𝑖̈−𝑠_𝑖𝜃̇²−2ℎ̇_𝑖𝜃̇−2𝑚₀ℎ̇₀𝜃̇=𝐹sin𝛿,(3.5)_𝑖=1𝑚𝑚_𝑖𝑎_𝑥𝑠_𝑖−ℎ_𝑖𝑠_𝑖̈+𝑠_𝑖ℎ̈_𝑖+2𝑠_𝑖𝑠_𝑖̇+ℎ_𝑖ℎ̇_𝑖𝜃̇+2𝑚₀ℎ₀ℎ̇₀𝜃̇₀+𝑚𝑏𝑎_𝑧̂=𝜏,(3.6)_𝑖𝑠^𝑖̈+𝑎_𝑧̂−ℎ_𝑖𝜃̈−𝑠_𝑖𝜃̇²−2ℎ̇_𝑖𝜃̇+𝑘_𝑖𝑠_𝑖+𝑐_𝑖𝑠_𝑖̇=0,∀𝑖,(3.7) where 𝑎_𝑧̂=𝑣̇_𝑧−𝜃̇𝑣_𝑥(𝑡). Here 𝑣_𝑥(𝑡) is considered as an exogenous input. The subsequent analysis is based on the above equations of motion for the transverse, pitch, and slosh dynamics of the vehicle.

Eliminating 𝑠_𝑖̈ in (3.5) and (3.6) using (3.7) yields 𝑚𝑚𝑏+−𝑚𝑚₀₀𝑎_𝑧̂+𝑚𝑏−𝑚₀ℎ₀𝜃̈−2𝑚₀ℎ̇₀𝜃̇−^𝑁_𝑖=1𝑘_𝑖𝑠_𝑖+𝑐_𝑖𝑠_𝑖̇=𝐹sin𝛿,ℎ₀𝑎_𝑧̂+𝐼−^𝑁_𝑖=1𝑚_𝑖ℎ^2𝑖𝜃̈+2𝑚⁰ℎ₀ℎ̇₀𝜃̇+𝐺=𝑀+𝐹𝑝sin𝛿,(3.8) where 𝐺=^𝑁_𝑖=1𝑚^𝑖𝑎_𝑥+𝑚_𝑖ℎ̈_𝑖+𝑘_𝑖ℎ_𝑖𝑠_𝑖+ℎ_𝑖𝑐_𝑖𝑠_𝑖̇+2𝑚_𝑖𝑠_𝑖𝑠_𝑖̇𝜃̇−𝑚_𝑖ℎ_𝑖𝑠_𝑖𝜃̇².(3.9) Note that the expressions (2.1) have been utilized to obtain (3.8) in the form above.

By defining control transformations from (𝛿,𝑀) to new control inputs (𝑢₁,𝑢₂): ⎡⎢⎢⎣𝑢₁𝑢₂⎤⎥⎥⎦=⎡⎢⎢⎢⎣𝑚+𝑚₀𝑚𝑏−𝑚₀ℎ₀𝑚𝑏−𝑚₀ℎ₀𝐼−∑^𝑁_𝑖=1𝑚_𝑖ℎ^2𝑖⎤⎥⎥⎥⎦⁻¹⎡⎢⎢⎢⎣𝐹sin𝛿+2𝑚₀ℎ̇₀𝜃̇+^𝑁𝑀+𝐹𝑝sin𝛿−2𝑚∑_𝑖=10𝑘_𝑖𝑠_𝑖+𝑐_𝑖𝑠_𝑖̇ℎ₀ℎ̇₀𝜃̇−𝐺⎤⎥⎥⎥⎦,(3.10) the system (3.5)–(3.7) can be written as 𝑣̇_𝑧=𝜃𝑢₁̈=+𝑢𝜃₂̇𝑣_𝑥(𝑡),(3.11)𝑠_𝑖̈,(3.12)=−𝜔^2𝑖𝑠_𝑖−2𝜁_𝑖𝜔_𝑖𝑠_𝑖̇−𝑢₁+ℎ_𝑖𝑢₂+𝑠_𝑖𝜃̇²+2ℎ̇_𝑖𝜃̇,∀𝑖,(3.13) where 𝜔^2𝑖=𝑘^𝑖𝑚_𝑖,2𝜁_𝑖𝜔_𝑖=𝑐^𝑖𝑚_𝑖,∀𝑖.(3.14) Here 𝜔_𝑖 and 𝜁_𝑖, for all 𝑖, denote the undamped natural frequencies and damping ratios, respectively.

The main idea in the subsequent development is to first design feedback control laws for (𝑢₁,𝑢₂) and then use the following equations to obtain the feedback laws for the original controls (𝛿,𝑀) for 𝑡≤𝑡_𝑏: 𝛿=sin⁻¹⎛⎜⎜⎝𝑚+𝑚₀𝑢₁+𝑚𝑏−𝑚₀ℎ₀𝑢₂−2𝑚₀ℎ̇₀𝜃̇−∑^𝑁𝑖=1𝑘_𝑖𝑠_𝑖+𝑐_𝑖𝑠_𝑖̇𝑀=𝑚𝑏−𝑚₀𝐹⎞⎟⎟⎠,(3.15)ℎ₀𝑢₁+𝐼−^𝑁_𝑖=1𝑚_𝑖ℎ^2𝑖𝑢²+2𝑚₀ℎ₀ℎ̇₀𝜃̇+𝐺−𝐹𝑝sin𝛿.(3.16)

Consider the following candidate Lyapunov function to stabilize the subsystem defined by (3.11) and (3.12): 𝑉=𝑟¹2𝑣^2𝑧+𝑟²2𝜃²+𝑟³2𝜃̇²,(3.17) where 𝑟₁,𝑟₂, and 𝑟₃ are positive constants so that the function 𝑉 is positive definite.

The time derivative of 𝑉 along the trajectories of (3.11) and (3.12) can be computed as 𝑉̇=𝑟₁𝑣_𝑧𝑣̇_𝑧+𝑟₂𝜃𝜃̇+𝑟₃𝜃̇𝜃̈(3.18) or rewritten in terms of the new control inputs 𝑉̇=𝑟₁𝑣_𝑧𝑢₁+𝑟₁𝑣_𝑥𝑣_𝑧+𝑟₂𝜃+𝑟₃𝑢₂𝜃̇.(3.19) Clearly, the feedback laws 𝑢𝑢₁=−𝑙₁𝑣_𝑧,(3.20)₂=−1𝑟₃𝑟₂𝜃+𝑙₂𝜃̇,(3.21) where 𝑙₁,𝑙₂ are positive constants and taking into account that 𝑟₁𝑣_𝑥𝑣_𝑧𝜃̇≤𝜃̇²2+𝑟¹𝑣_𝑥𝑣_𝑧²2,(3.22) yield 𝑉̇≤=−−𝑙𝑟₁₁𝑟₁𝑣^2𝑧−𝑙₂𝜃̇²+𝑟₁𝑣_𝑥𝑣_𝑧𝜃̇𝑙¹−𝑟¹𝑣^2𝑥2𝑣^2𝑧−𝑙₂−12𝜃̇²,(3.23) which satisfies 𝑉̇≤0 if 𝑙₁>0.5𝑟₁𝑣^2𝑥 and 𝑙₂>0.5.

The closed-loop system for (𝑣_𝑧,𝜃)-dynamics can be written as 𝑣𝜃̇_𝑧̈==−−𝐾𝑙₁𝑣_𝑧1+𝜃̇𝑣_𝑥(𝑡),(3.24)𝜃−𝐾₂𝜃̇,(3.25) where 𝐾₁=𝑟₂/𝑟₃ and 𝐾₂=𝑙₂/𝑟₃.

Equation (3.25) can be easily solved in the case of 𝐾²²>4𝐾₁ as 𝜃(𝑡)=𝐴𝑒^−𝜆1𝑡+𝐵𝑒^−𝜆2𝑡,(3.26) where 𝐴,𝐵 are integration constants and −𝜆₁,−𝜆₂ are the eigenvalues of the linear system (3.25). Therefore, 𝜃(𝑡) and 𝜃̇(𝑡) can be upper bounded as ||𝜃(𝑡)||≤𝐶𝑒^−𝜆𝑡,||𝜃̇(𝑡)||≤𝐷𝑒^−𝜆𝑡,(3.27) respectively, where 𝐶,𝐷 are positive constants and 𝜆=min(𝜆₁,𝜆₂). Now, assuming that 𝜆≠𝑙₁, (3.24) can be integrated to obtain an upper bound for 𝑣_𝑧(𝑡) as ||𝑣_𝑧(𝑡)||≤𝛼𝑒^−𝛽𝑡,(3.28) where 𝛼,𝛽 are positive constants. Therefore, it can be concluded that the (𝑣_𝑧,𝜃)-dynamics are exponentially stable under the control laws (3.20) and (3.21).

To analyze the stability of the 𝑁 equations defined by (3.13), it will be first shown that the system described by the equation 𝑠_𝑖̈+2𝜁_𝑖𝜔_𝑖(𝑡)𝑠_𝑖̇+𝜔^2𝑖(𝑡)𝑠_𝑖=0(3.29) is exponentially stable.

From (2.4) and (2.6), 𝜔_𝑖(𝑡)=2𝑔𝜉_𝑖𝜑tanh2𝜉^𝑖ℎ(𝑡)𝜑∈𝐶¹.(3.30) The following properties can be shown to hold: 𝜔^2𝑖(𝑡)≥𝜀²¹,𝑝(𝑡)=12𝜔̇^𝑖(𝑡)𝜔_𝑖(𝑡)+2𝜁^𝑖||2𝜁_𝑖𝜔_𝑖(𝑡)≥𝜀²²,𝜔_𝑖(𝑡)||≤2𝜁_𝑖2𝑔𝜉_𝑖𝜑=𝑀¹,||𝜔^2𝑖(𝑡)||≤2𝑔𝜉^𝑖𝜑=𝑀²||2𝜔̇_𝑖,(𝑡)𝜔_𝑖(𝑡)||≤𝑔2𝜉^𝑖𝜑²=𝑀₃,(3.31) where 𝜀₁ and 𝜀₂ are small positive parameters given the fact that the tank will never be totally empty, but a small amount of fuel will always remain inside. For this same reason, ℎ(𝑡)>0, for all 𝑡. Therefore, by Corollary A.2 in the Appendix, the system (3.29) is exponentially stable.

Now write (3.13) as 𝑥̇=𝐴₁(𝑡)+𝐴₂(𝑡)𝑥+𝐻(𝑡),(3.32) where 𝑥=[𝑠_𝑖,𝑠_𝑖̇]^𝑇 and 𝐴₁(𝑡)=⎡⎢⎢⎣−𝜔0_2𝑖1(𝑡)−2𝜁_𝑖𝜔_𝑖(𝑡)⎤⎥⎥⎦,𝐴²(𝑡)=⎡⎢⎢⎣𝜃̇00₂𝐻(𝑡)=⎡⎢⎢⎣−𝑎_𝑧̂0(𝑡)0⎤⎥⎥⎦,(𝑡)+ℎ_𝑖(𝑡)𝜃̈(𝑡)+2ℎ̇_𝑖(𝑡)𝜃̇(𝑡)⎤⎥⎥⎦.(3.33)

Under the stated assumptions, 𝐴₁(𝑡) is exponentially stable (see the Appendix) and there exist positive constants 𝜆₀,𝜆₁, and 𝜆₂ such that ^∞0‖‖𝐴₂(𝑡)‖‖𝑑𝑡≤𝜆₀,‖𝐻(𝑡)‖≤𝜆₁𝑒^−𝜆2𝑡,∀𝑡≥0.(3.34) Hence, for any initial condition, the state of the system (3.5)–(3.7) converges exponentially to zero.

4. Simulation

The feedback control law developed in the previous section is implemented here for the fourth stage of the European launcher Vega. The first two slosh modes are included to demonstrate the effectiveness of the controller (3.15), (3.16), (3.20), (3.21) by applying to the complete nonlinear system (2.20)–(2.23). The physical parameters used in the simulations are given in Table 1.

Table 1 

Physical parameters for AVUM stage of Vega.

We consider stabilization of the spacecraft in orbital transfer, suppressing the transverse and pitching motion of the spacecraft and sloshing of fuel while the spacecraft is accelerating. In other words, the control objective is to stabilize the relative equilibrium corresponding to a specific spacecraft axial acceleration and 𝑣_𝑧=𝜃=𝜃̇=𝑠_𝑖=𝑠_𝑖̇=0,𝑖=1,2.

Time responses shown in Figures 2, 3, and 4 correspond to the initial conditions 𝑣_𝑥0=3000 m/s, 𝑣_𝑧0=100 m/s, 𝜃₀=5^∘, 𝜃̇₀=0, 𝑠₁₀=0.1 m, 𝑠₂₀=−0.1 m, and 𝑠̇₁₀=𝑠̇₂₀=0. We assume a fuel burn time of 650 seconds. As can be seen, the transverse velocity, attitude angle, and the slosh states converge to the relative equilibrium at zero while the axial velocity 𝑣_𝑥 increases and 𝑣̇_𝑥 tends asymptotically to 𝐹/(𝑚+𝑚_𝑓). Note that there is a trade-off between good responses for the directly actuated degrees of freedom (the transverse and pitch dynamics) and good responses for the internal degrees of freedom (the slosh dynamics); the controller given by (3.15), (3.16), (3.20), (3.21) with parameters 𝑟₁=8×10⁻⁷, 𝑟₂=10³, 𝑟₃=500, 𝑙₁=10⁴, and 𝑙₂=4×10⁴ represents one example of this balance.

(a)

(b)

(c)

Figure 2 

Time responses of 𝑣𝑥,𝑣𝑧, and 𝜃.

Figure 3 

Time responses of 𝑠1 and 𝑠2.

Figure 4 

Gimbal deflection angle 𝛿 and pitching moment 𝑀.

Figures 5, 6, and 7 show the results of a simulation with no control (𝑀=𝛿=0) using the same initial conditions and physical parameters as above. As expected, the fuel slosh dynamics destabilize the uncontrolled spacecraft.

Figure 5 

Time responses of 𝑣𝑥,𝑣𝑧 and 𝜃 (zero control case).

Figure 6 

Time responses of 𝑠1 and 𝑠2 (zero control case).

Figure 7 

Gimbal deflection angle 𝛿 and pitching moment 𝑀 (zero control case).

5. Conclusions

A complete nonlinear dynamical model has been developed for a spacecraft with multiple slosh modes that have time-varying parameters. A feedback controller has been designed to achieve stabilization of the pitch and transverse dynamics as well as suppression of the slosh modes, while the spacecraft accelerates in the axial direction. The effectiveness of the feedback controller has been illustrated through a simulation example.

The many avenues considered for future research include problems involving multiple liquid containers and three-dimensional transfers. Future research also includes designing nonlinear observers to estimate the slosh states as well as nonlinear control laws that achieve robustness, insensitivity to system and control parameters, and improved disturbance rejection.

Appendix

Consider the system𝑠̈+𝑓(𝑡)𝑠̇+𝑔(𝑡)𝑠=0,(A.1) where 𝑔(𝑡)∈𝐶¹, |𝑓(𝑡)|<𝑀₁, |𝑔(𝑡)|<𝑀₂, |𝑔̇(𝑡)|<𝑀₃.

Theorem A.1. If 𝑔(𝑡)>𝜀²¹ and 𝑝(𝑡)=(1/2)(𝑔̇(𝑡)/𝑔(𝑡))+𝑓(𝑡)>𝜀²², then the origin is globally uniformly asymptotically stable.

Proof. Given the conditions above, the following bounds can be set: −−𝑀𝑀₁₃<𝑓(𝑡)<𝑀₁,𝛼²¹<𝑔(𝑡)<𝑀₂,<𝑔̇(𝑡)<𝑀₃,𝜀²²<𝑝(𝑡)<𝑀³2𝜀²¹+𝑀₁.(A.2)
Consider the following candidate Lyapunov function: 𝑉(𝑧,𝑡)=12𝑠²+2𝛽𝑠𝑠̇√𝑔(𝑡)+𝑔𝑠(²̇𝑡),(A.3) where 𝑧=[𝑠𝑠]̇^𝑇 is the state vector and 𝛽 is a positive constant. This function can be rewritten in a matrix form as 𝑉(𝑧,𝑡)=12𝑠𝑠̇⎡⎢⎢⎢⎢⎣1𝛽𝛽√𝑔√𝑔𝑔1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎣𝑠𝑠⎤⎥⎥⎦̇,(A.4) which is positive definite if 𝛽<1.
Recalling that a positive definite quadratic function 𝑧^𝑇𝑃𝑧 satisfies 𝜆_min(𝑃)𝑧^𝑇𝑧≤𝑧^𝑇𝑃𝑧≤𝜆_max(𝑃)𝑧^𝑇𝑧,(A.5) where 𝜆_min(𝑃)=1+𝑔2𝑔⎡⎢⎢⎣1−1−4𝑔1−𝛽²(1+𝑔)²𝜆_max⎤⎥⎥⎦,(𝑃)=1+𝑔2𝑔⎡⎢⎢⎣1+1−4𝑔1−𝛽²(1+𝑔)²⎤⎥⎥⎦,(A.6) and thus the following hold: 𝛾₁‖𝑧‖²≤𝑉≤𝛾₂‖𝑧‖²,(A.7) where 𝛾₁ and 𝛾₂ are positive constants.
Taking the time derivative of 𝑉(𝑧,𝑡) yields 𝑉̇=−𝛽√𝑔(𝑡)𝑔(𝑡)𝑠²+𝑝(𝑡)𝑠𝑠̇+𝑝(𝑡)𝛽√𝑔(𝑡)−1𝑠²̇,(A.8) which can be rewritten as 𝑉̇=−𝛽√𝑔𝑠𝑠̇⎡⎢⎣⎢⎢𝑔𝑝𝑝22𝛽√𝑝𝑔−1⎤⎥⎥⎥⎦⎡⎢⎢⎣𝑠𝑠⎤⎥⎥⎦̇<0.(A.9) Clearly, 𝑉̇<0 if 𝛽<16𝜀⁵¹𝜀²²16𝑀₂𝜀⁴¹+𝑀₃+2𝜀²¹𝑀₁².(A.10) Note that 𝑉̇ satisfies 𝑉̇≤−𝛽√𝑔𝜆^min(𝑄)‖𝑧‖².(A.11) It can be shown that if 𝛽<min1,𝜀²²1−𝑀₂√𝑀₂,16𝜀⁵¹𝜀²²16𝑀₂𝜀⁴¹+𝑀₃+2𝜀²¹𝑀₁²,(A.12) then, using Theorem 4.10 of [24], it can be concluded that the origin is exponentially stable. Hence, the following result can be stated.

Corollary A.2. There exist 𝛼,𝛽>0 such that |𝑠|<𝛽𝑒^{−𝛼(𝑡−𝑡0)},|𝑠|̇<𝛽𝑒^{−𝛼(𝑡−𝑡0)},∀𝑡≥𝑡₀.(A.13)
The following result is a modified version of that presented in [20].

Lemma A.3. Consider a system that is described by the linear time-varying differential equation 𝑥̇=𝐴₁(𝑡)+𝐴₂(𝑡)𝑥+𝐻(𝑡),𝑥∈ℝ^𝑛.(A.14)
If the matrix 𝐴₁(𝑡) is exponentially stable and there exist positive constants 𝜆₀, 𝜆₁, and 𝜆₂ such that (i)^∞0‖‖𝐴₂(𝑡)‖‖𝑑𝑡≤𝜆₀,(ii)‖𝐻(𝑡)‖≤𝜆₁𝑒^−𝜆2𝑡,∀𝑡≥0,(A.15) then all the solutions of (A.14) approach zero exponentially as 𝑡 goes to ∞.

Acknowledgment

The authors wish to acknowledge the support provided by Embry-Riddle Aeronautical University.